Asset Returns and Moments of Asset Returns Using Portfolio Object

Since mean-variance portfolio optimization problems require estimates for the mean and covariance of asset returns, the Portfolio object has several ways to set and get the properties AssetMean (for the mean) and AssetCovar (for the covariance). In addition, the return for a riskless asset is kept in the property RiskFreeRate so that all assets in AssetMean and AssetCovar are risky assets. For information on the workflow when using Portfolio objects, see Portfolio Object Workflow.

Assignment Using the Portfolio Function

Suppose that you have a mean and covariance of asset returns in variables m and C. The properties for the moments of asset returns are set using the Portfolio object:

m = [ 0.05; 0.1; 0.12; 0.18 ];
C = [ 0.0064 0.00408 0.00192 0; 
    0.00408 0.0289 0.0204 0.0119;
    0.00192 0.0204 0.0576 0.0336;
    0 0.0119 0.0336 0.1225 ];
m = m/12;
C = C/12;
p = Portfolio('AssetMean', m, 'AssetCovar', C);
disp(p.NumAssets)
disp(p.AssetMean)
disp(p.AssetCovar)

      4

    0.0042
    0.0083
    0.0100
    0.0150

    0.0005    0.0003    0.0002         0
    0.0003    0.0024    0.0017    0.0010
    0.0002    0.0017    0.0048    0.0028
         0    0.0010    0.0028    0.0102

Notice that the Portfolio object determines the number of assets in NumAssets from the moments. The Portfolio object enables separate initialization of the moments, for example:

m = [ 0.05; 0.1; 0.12; 0.18 ];
C = [ 0.0064 0.00408 0.00192 0; 
      0.00408 0.0289 0.0204 0.0119;
      0.00192 0.0204 0.0576 0.0336;
      0 0.0119 0.0336 0.1225 ];
m = m/12;
C = C/12;
 
p = Portfolio;
p = Portfolio(p, 'AssetMean', m);
p = Portfolio(p, 'AssetCovar', C);
[assetmean, assetcovar] = p.getAssetMoments

assetmean =

    0.0042
    0.0083
    0.0100
    0.0150


assetcovar =

    0.0005    0.0003    0.0002         0
    0.0003    0.0024    0.0017    0.0010
    0.0002    0.0017    0.0048    0.0028
         0    0.0010    0.0028    0.0102

The getAssetMoments function lets you get the values for AssetMean and AssetCovar properties at the same time.

Assignment Using the `setAssetMoments` Function

You can also set asset moment properties using the setAssetMoments function. For example, given the mean and covariance of asset returns in the variables m and C, the asset moment properties can be set:

m = [ 0.05; 0.1; 0.12; 0.18 ];
C = [ 0.0064 0.00408 0.00192 0; 
    0.00408 0.0289 0.0204 0.0119;
    0.00192 0.0204 0.0576 0.0336;
    0 0.0119 0.0336 0.1225 ];
m = m/12;
C = C/12;
 
p = Portfolio;
p = setAssetMoments(p, m, C);
[assetmean, assetcovar] = getAssetMoments(p)

assetmean =

    0.0042
    0.0083
    0.0100
    0.0150

assetcovar =

    0.0005    0.0003    0.0002         0
    0.0003    0.0024    0.0017    0.0010
    0.0002    0.0017    0.0048    0.0028
         0    0.0010    0.0028    0.0102

Scalar Expansion of Arguments

Both the Portfolio object and the setAssetMoments function perform scalar expansion on arguments for the moments of asset returns. When using the Portfolio object, the number of assets must be already specified in the variable NumAssets. If NumAssets has not already been set, a scalar argument is interpreted as a scalar with NumAssets set to 1. setAssetMoments provides an additional optional argument to specify the number of assets so that scalar expansion works with the correct number of assets. In addition, if either a scalar or vector is input for the covariance of asset returns, a diagonal matrix is formed such that a scalar expands along the diagonal and a vector becomes the diagonal. This example demonstrates scalar expansion for four jointly independent assets with a common mean 0.1 and common variance 0.03:

p = Portfolio;
p = setAssetMoments(p, 0.1, 0.03, 4);
[assetmean, assetcovar] = getAssetMoments(p)

assetmean =

    0.1000
    0.1000
    0.1000
    0.1000

assetcovar =

    0.0300         0         0         0
         0    0.0300         0         0
         0         0    0.0300         0
         0         0         0    0.0300

If at least one argument is properly dimensioned, you do not need to include the additional NumAssets argument. This example illustrates a constant-diagonal covariance matrix and a mean of asset returns for four assets:

p = Portfolio;
p = setAssetMoments(p, [ 0.05; 0.06; 0.04; 0.03 ], 0.03);
[assetmean, assetcovar] = getAssetMoments(p)

assetmean =

    0.0500
    0.0600
    0.0400
    0.0300

assetcovar =

    0.0300         0         0         0
         0    0.0300         0         0
         0         0    0.0300         0
         0         0         0    0.0300

In addition, scalar expansion works with the Portfolio object if NumAssets is known, or is deduced from the inputs.

Estimating Asset Moments from Prices or Returns

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This example shows how to set the moments of asset returns by using the estimateAssetMoments function which accepts either prices or returns and estimates the mean and covariance of asset returns. Either prices or returns are stored as matrices with samples going down the rows and assets going across the columns. In addition, prices or returns can be stored in a table or timetable (see Estimating Asset Moments from Time Series Data).

To illustrate using estimateAssetMoments, generate random samples of 120 observations of asset returns for four assets from the mean and covariance of asset returns in the variables m and C with portsim. The default behavior of portsim creates simulated data with estimated mean and covariance identical to the input moments m and C. In addition to a return series created by portsim in the variable X, a price series is created in the variable Y.

m = [ 0.05; 0.1; 0.12; 0.18 ];
C = [ 0.0064 0.00408 0.00192 0; 
      0.00408 0.0289 0.0204 0.0119;
      0.00192 0.0204 0.0576 0.0336;
      0 0.0119 0.0336 0.1225 ];
m = m/12;
C = C/12;
X = portsim(m', C, 120);
Y = ret2tick(X);

Note that Portfolio optimization requires that you use total returns and not just price returns. So, "returns" should be total returns and "prices" should be total return prices.

Given asset returns and prices in variables X and Y, the following sequence of code demonstrates equivalent ways to estimate asset moments for the Portfolio object. A Portfolio object is created in p with the moments of asset returns set directly in the Portfolio object, and a second Portfolio object is created in q to obtain the mean and covariance of asset returns from asset return data in X using estimateAssetMoments.

m = [ 0.05; 0.1; 0.12; 0.18 ];
C = [ 0.0064 0.00408 0.00192 0; 
      0.00408 0.0289 0.0204 0.0119;
      0.00192 0.0204 0.0576 0.0336;
      0 0.0119 0.0336 0.1225 ];
m = m/12;
C = C/12;
 
X = portsim(m', C, 120);
p = Portfolio('mean', m, 'covar', C);
q = Portfolio;
q = estimateAssetMoments(q, X);
 
[passetmean, passetcovar] = getAssetMoments(p)

passetmean = 4×1

    0.0042
    0.0083
    0.0100
    0.0150

passetcovar = 4×4

    0.0005    0.0003    0.0002         0
    0.0003    0.0024    0.0017    0.0010
    0.0002    0.0017    0.0048    0.0028
         0    0.0010    0.0028    0.0102

[qassetmean, qassetcovar] = getAssetMoments(q)

qassetmean = 4×1

    0.0042
    0.0083
    0.0100
    0.0150

qassetcovar = 4×4

    0.0005    0.0003    0.0002    0.0000
    0.0003    0.0024    0.0017    0.0010
    0.0002    0.0017    0.0048    0.0028
    0.0000    0.0010    0.0028    0.0102

Notice how either approach has the same moments. The default behavior of estimateAssetMoments is to work with asset returns. If, instead, you have asset prices in the variable Y, estimateAssetMoments accepts a name-value pair argument name 'DataFormat' with a corresponding value set to 'prices' to indicate that the input to the function is in the form of asset prices and not returns (the default value for the 'DataFormat' argument is 'returns'). The following code compares direct assignment of moments in the Portfolio object p with estimated moments from asset price data in Y in the Portfolio object q.

m = [ 0.05; 0.1; 0.12; 0.18 ];
C = [ 0.0064 0.00408 0.00192 0; 
      0.00408 0.0289 0.0204 0.0119;
      0.00192 0.0204 0.0576 0.0336;
      0 0.0119 0.0336 0.1225 ];
m = m/12;
C = C/12;
 
X = portsim(m', C, 120);
Y = ret2tick(X);

p = Portfolio('mean',m,'covar',C);
        
q = Portfolio;
q = estimateAssetMoments(q, Y, 'dataformat', 'prices');
 
[passetmean, passetcovar] = getAssetMoments(p)

passetmean = 4×1

    0.0042
    0.0083
    0.0100
    0.0150

passetcovar = 4×4

    0.0005    0.0003    0.0002         0
    0.0003    0.0024    0.0017    0.0010
    0.0002    0.0017    0.0048    0.0028
         0    0.0010    0.0028    0.0102

[qassetmean, qassetcovar] = getAssetMoments(q)

qassetmean = 4×1

    0.0042
    0.0083
    0.0100
    0.0150

qassetcovar = 4×4

    0.0005    0.0003    0.0002    0.0000
    0.0003    0.0024    0.0017    0.0010
    0.0002    0.0017    0.0048    0.0028
    0.0000    0.0010    0.0028    0.0102

Estimating Asset Moments with Missing Data

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This example shows how to work with missing data indicated by NaN values in your return or price data. Although Multivariate Normal Regression goes into detail about regression with missing data, the estimateAssetMoments function has a name-value pair argument name 'MissingData' that indicates with a Boolean value whether to use the missing data capabilities of Financial Toolbox™ software. The default value for 'MissingData' is false which removes all samples with NaN values. If, however, 'MissingData' is set to true, estimateAssetMoments uses the ECM algorithm to estimate asset moments. The following code illustrates how this works on price data with missing values:

m = [ 0.05; 0.1; 0.12; 0.18 ];
C = [ 0.0064 0.00408 0.00192 0; 
      0.00408 0.0289 0.0204 0.0119;
      0.00192 0.0204 0.0576 0.0336;
      0 0.0119 0.0336 0.1225 ];
m = m/12;
C = C/12;
 
X = portsim(m', C, 120);
Y = ret2tick(X);
Y(1:20,1) = NaN;
Y(1:12,4) = NaN;

p = Portfolio('mean',m,'covar',C);
        
q = Portfolio;
q = estimateAssetMoments(q, Y, 'dataformat', 'prices');
 
r = Portfolio;
r = estimateAssetMoments(r, Y, 'dataformat', 'prices', 'missingdata', true);

[passetmean, passetcovar] = getAssetMoments(p)

passetmean = 4×1

    0.0042
    0.0083
    0.0100
    0.0150

passetcovar = 4×4

    0.0005    0.0003    0.0002         0
    0.0003    0.0024    0.0017    0.0010
    0.0002    0.0017    0.0048    0.0028
         0    0.0010    0.0028    0.0102

[qassetmean, qassetcovar] = getAssetMoments(q)

qassetmean = 4×1

    0.0039
    0.0128
    0.0153
    0.0238

qassetcovar = 4×4

    0.0006    0.0004    0.0001    0.0001
    0.0004    0.0023    0.0015    0.0005
    0.0001    0.0015    0.0043    0.0024
    0.0001    0.0005    0.0024    0.0084

[rassetmean, rassetcovar] = getAssetMoments(r)

rassetmean = 4×1

    0.0033
    0.0083
    0.0100
    0.0176

rassetcovar = 4×4

    0.0006    0.0004    0.0001    0.0001
    0.0004    0.0024    0.0017    0.0007
    0.0001    0.0017    0.0048    0.0025
    0.0001    0.0007    0.0025    0.0083

The Portfolio object p contains raw moments, the Portfolio object q contains estimated moments in which NaN values are discarded, and the Portfolio object r contains raw moments that accommodate missing values. Each time you run this code, you will get different estimates for the moments in q and r, and these will also differ from the moments in p.

Estimating Asset Moments from Time Series Data

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This example shows how the estimateAssetMoments function also accepts asset returns or prices stored in a table or timetable. estimateAssetMoments implicitly works with matrices of data or data in a table or timetable object using the same rules for whether the data are returns or prices.

To illustrate the use of a table and timetable, use array2table and array2timetable to create a table and a timetable that contain asset returns generated with portsim (see Estimating Asset Moments from Prices or Returns). Two portfolio objects are then created with the AssetReturns based on a table and a timetable object.

m = [ 0.05; 0.1; 0.12; 0.18 ];
C = [ 0.0064 0.00408 0.00192 0; 
      0.00408 0.0289 0.0204 0.0119;
      0.00192 0.0204 0.0576 0.0336;
      0 0.0119 0.0336 0.1225 ];
m = m/12;
C = C/12;
 
assetRetnScenarios = portsim(m', C, 120);
dates = datetime(datenum(2001,1:120,31), 'ConvertFrom', 'datenum');
assetsName = {'Bonds', 'LargeCap', 'SmallCap', 'Emerging'};
assetRetnTimeTable = array2timetable(assetRetnScenarios,'RowTimes',dates, 'VariableNames', assetsName);      
assetRetnTable = array2table(assetRetnScenarios, 'VariableNames', assetsName);


% Create two Portfolio objects:
% p with predefined mean and covar and q with asset return scenarios to estimate mean and covar.
p = Portfolio('mean', m, 'covar', C);
q = Portfolio;

% Estimate asset moments using a timetable
q = estimateAssetMoments(q, assetRetnTimeTable);
[passetmean, passetcovar] = getAssetMoments(p)

passetmean = 4×1

    0.0042
    0.0083
    0.0100
    0.0150

passetcovar = 4×4

    0.0005    0.0003    0.0002         0
    0.0003    0.0024    0.0017    0.0010
    0.0002    0.0017    0.0048    0.0028
         0    0.0010    0.0028    0.0102

[qassetmean, qassetcovar] = getAssetMoments(q)

qassetmean = 4×1

    0.0042
    0.0083
    0.0100
    0.0150

qassetcovar = 4×4

    0.0005    0.0003    0.0002    0.0000
    0.0003    0.0024    0.0017    0.0010
    0.0002    0.0017    0.0048    0.0028
    0.0000    0.0010    0.0028    0.0102

% Estimate asset moments using a table
q = estimateAssetMoments(q, assetRetnTable);
[passetmean, passetcovar] = getAssetMoments(p)

passetmean = 4×1

    0.0042
    0.0083
    0.0100
    0.0150

passetcovar = 4×4

    0.0005    0.0003    0.0002         0
    0.0003    0.0024    0.0017    0.0010
    0.0002    0.0017    0.0048    0.0028
         0    0.0010    0.0028    0.0102

[qassetmean, qassetcovar] = getAssetMoments(q)

qassetmean = 4×1

    0.0042
    0.0083
    0.0100
    0.0150

qassetcovar = 4×4

    0.0005    0.0003    0.0002    0.0000
    0.0003    0.0024    0.0017    0.0010
    0.0002    0.0017    0.0048    0.0028
    0.0000    0.0010    0.0028    0.0102

As you can see, the moments match between th two portfolios. In addition, estimateAssetMoments also extracts asset names or identifiers from a table or timetable when the argument name 'GetAssetList' is set to true (its default value is false). If the 'GetAssetList' value is true, the identifiers are used to set the AssetList property of the Portfolio object. To show this, the formation of the Portfolio object q is repeated from the previous code with the 'GetAssetList' flag set to true and this extracts the column labels from a table or timetable object.

q = estimateAssetMoments(q,assetRetnTable,'GetAssetList',true);
disp(q.AssetList)

    {'Bonds'}    {'LargeCap'}    {'SmallCap'}    {'Emerging'}

Note if you set the 'GetAssetList' flag set to true and your input data is in a matrix, estimateAssetMoments uses the default labeling scheme from setAssetList described in Setting Up a List of Asset Identifiers.